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An analysis of random elections with large numbers of voters

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  • Harrison-Trainor, Matthew

Abstract

In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate A defeats candidate B, B defeats C, and C defeats A. It is known that such cycles are unlikely to occur. Under the Impartial Culture assumption, in a random election with three candidates and a very large number of voters there is a 91.23% chance of avoiding a cycle. By studying the geometry of the space of random elections, we give a mathematical explanation of why this is the case. Our main result is that margin graphs that are more cyclic in a certain precise sense are less likely to occur. This connects the probabilistic study of voting methods to Zwicker’s analysis of Condorcet’s paradox in terms of cycles and cuts.

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  • Harrison-Trainor, Matthew, 2022. "An analysis of random elections with large numbers of voters," Mathematical Social Sciences, Elsevier, vol. 116(C), pages 68-84.
    • Handle: RePEc:eee:matsoc:v:116:y:2022:i:c:p:68-84
    DOI: 10.1016/j.mathsocsci.2022.01.002
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    Cited by:

    1. Holliday, Wesley H., 2024. "An impossibility theorem concerning positive involvement in voting," Economics Letters, Elsevier, vol. 236(C).
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    3. Wesley H. Holliday, 2024. "An impossibility theorem concerning positive involvement in voting," Papers 2401.05657, arXiv.org, revised Feb 2024.

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